Cube Handling in Races

Forum Archive : Cube Handling in Races

Lamford's race forumla

From:   Michael Schell
Date:   19 August 2001
Subject:   Lamford's race forumla
Google:   YSXf7.1579$

One of the things Paul Lamford offers in his new book ("Starting Out in
Backgammon") is a formula for doubling in non-contact races. It's
essentially an alternative to the similar schemes from Thorp, Ward, et al.
Lamford suggests that the system is highly reliable down to 50 pips,
increasingly less reliable to 20 pips, and generally unreliable below 20

Lamford's system is as follows:

1. Get the base pip count
2. Add  point for each extra crossover
3. Adjust for "wastage" as follows:
   a. Add two pips for each checker over two on the one-point
   b. Add one pip for each checker over two on the two-point
   c. Add .5 pip for each checker over five on any point
4. Adjust for gaps as follows IF there are at least six checkers above the
   a. Add four pips for a gap on the four-point
   b. Add three pips for a gap on the five-point
   c. Add two pips for a gap on the three-point
   d. Add one point for a gap on the two-point
5. Make your decision:
   a. Double if you lead in the adjusted pip count by at least 10% (of
      *your* pip count)
   b. Redouble if you lead in the adjusted pip count by at least 11%
   c. Take if you trail by no more than 12.5% (of opponent's pip count)

It would be interesting to hear opinions about how this system compares to
the other race formulae out there, both for accuracy and ease of use.

One problem I'm having learning this system is resolving the details of its
application. Lamford's explanation in "Starting Out in Backgammon" leaves
room for different interpretations. I'll go over a few questions I have
about applying the rules.


"Add  point for each extra crossover"

It's clear from Lamford's text that "extra crossover" means the DIFFERENCE
in crossovers between the side with the most crossovers and the side with
the fewest crossovers. So, if you have ten crossovers while your opponent
has eight, you'd add one point to your pip count (not five to yours and
four to your opponent's). For example, here is Lamford's Diagram 47:

    Diagram 47
    Money game.

    O on roll. Cube action?
    |    X  X  X  X  X |   |  X  X  X  X       |
    |    X  X          |   |  X  X  X          |
    |                  |   |  X                |
    |                  |   |                   |
    |                  |   |                   |
    |                  |BAR|                   |
    |                  |   |                   |
    |                  |   |  O                |
    |                  |   |  O                |
    |          O  O  O |   |  O  O             |
    |       O  O  O  O |   |  O  O  O        O |
    Pipcount  X: 105  O: 97
    CubeValue:  1

There is no adjustment made for extra crossovers, since O and X both have
seven crossovers remaining in the bear-in. Lamford gives this as a straight
97-105, making it No Double/Take (but see the discussion of this position
below under "Gaps").

Another ambiguity though is: what exactly does Lamford mean by "crossover"?
In the text he says "The board comprises four quadrants of six points each
and the move of a checker from one quadrant of the board into another is
known as a crossover". Clearly he doesn't count bearing off a checker as a
crossover (as evidenced by Diagram 47). And presumably, if you have any
checkers in your opponent's outer board, this would be considered two
crossovers, worth a one point penalty if you're the side with the higher
number of crossovers. But none of Lamford's examples of the adjusted pip
count show either side with checkers more than six pips away from the home
board, so this is not totally clear.


Wastage adjustments, along with gaps adjustments, account for poor checker
distribution in the home board. It appears from the text that wastage
adjustments are ALWAYS included in an adjusted pip count, regardless of the
stage of the race. Now take another look at the rules for applying the
adjustments, and tell me what you think should be done with the sixth
checker on the one- or two-point:

   a. Add two pips for each checker over two on the one-point
   b. Add one pip for each checker over two on the two-point
   c. Add  pip for each checker over five on any point

Do you add 2.5 pips for the sixth checker on the one-point (combining rules
a. and c.), or do you add just 2 pips (using only rule a.)? Unfortunately,
none of the positions in the book have more than five checkers on the one-
or two-point, so it's not clear from Lamford's examples.


The question here is: how soon in the race do you apply the adjustments for
home board gaps? Presumably you'd adjust for gaps once all your checkers
are borne in. But in Lamford's Diagram 44 (below), he doesn't adjust for
either player's gaps, even though O has no outfield checkers.

    Diagram 44
    Money game.

    O on roll. Cube action?
    |             X  X |   |  X  X  X          |
    |             X  X |   |  X  X  X          |
    |                X |   |  X  X  X          |
    |                X |   |                   |
    |                  |   |                   |
    |                  |BAR| (9)               |
    |                  |   |  O                |
    |                  |   |  O                |
    |                  |   |  O  O             |
    |                  |   |  O  O  O          |
    |                  |   |  O  O  O  O       |
    Pipcount  X: 89  O: 80
    CubeValue:  1

Here Lamford calculates the adjusted pip count as 82-92 (penalizing O two
pips for the six-point stack, and X three points for the extra crossovers),
making this a (Re)Double/Take. Adding one point for O's two-point gap would
make it 83-92 (still a Double, but not a Redouble), while adjusting for O
*and* X's gaps would make it 83-95 (Redouble/Pass). Lamford also doesn't
adjust for gaps in Diagram 47 (above), where both sides still have seven
checkers remaining to bear in.

But here is Lamford's Diagram 48:

    Diagram 48
    Money game.

    O on roll. Cube action?
    |             X  X |   |  X  X     X  X  X |
    |                X |   |  X        X  X  X |
    |                X |   |                 X |
    |                  |   |                 X |
    |                  |   |                   |
    |                  |BAR|                   |
    |                  |   |                   |
    |                  |   |  O                |
    |                  |   |  O  O  O          |
    |                  |   |  O  O  O  O  O    |
    |                  |   |  O  O  O  O  O  O |
    Pipcount  X: 60  O: 62
    CubeValue:  1

In this case, Lamford penalizes X four pips for the four-point gap, even
though X will likely have two rolls in which to try to cover it.
Additionally, X gets dinged for four extra crossovers and the extra two
checkers on the one-point, making the adjusted pip count 62-70, or

An interesting point with this last position is that all the traditional
race formulae (Thorp, Ward, Kleinmann, Robertie) blow this position. Snowie
full rollouts confirm Double/Pass here. (Actually Snowie full rollouts
agree with all of Lamford's assessments in these positions.) On the other
hand, taking out the gaps adjustment would make this 62-66 or No
Double/Take! So it's often critical to know when to apply this adjustment.
Unfortunately, though, there is no explanation forthcoming about why the
gap penalty is applied in Diagram 48, but not in Diagram 44 or 47.

Perhaps someone better acquainted with Lamford's system can help clarify
these procedural questions. At any rate, his is an interesting book, that
seems to take current theory and bot-derived knowledge into account. I've
done extensive rollouts on the diagrams in the book, and there are very few
positions where the rollout results differ substantially from Lamford's

- Michael Schell
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Cube Handling in Races

Bower's modified Thorp count  (Chuck Bower, July 1997) 
Calculating winning chances  (Raccoon, Jan 2007) 
Calculating winning chances  (OpenWheel+, Nov 2005)  [GammOnLine forum]
Doubling formulas  (Michael J. Zehr, Jan 1995) 
Doubling in a long race  (Brian Sheppard, Feb 1998) 
EPC example  (adambulldog+, Jan 2011) 
EPC example: stack and straggler  (neilkaz+, Jan 2009) 
EPC examples: stack and straggler  (Carlo Melzi+, Dec 2008) 
Effective pipcount  (Douglas Zare, Sept 2003)  [GammOnLine forum]
Effective pipcount and type of position  (Douglas Zare, Jan 2004)  [GammOnLine forum]
Kleinman count  (Øystein Johansen+, Feb 2001) 
Kleinman count  (André Nicoulin, Sept 1998) 
Kleinman count  (Chuck Bower, Mar 1998)  [Recommended reading]
Lamford's race forumla  (Michael Schell, Aug 2001) 
N-roll vs n-roll bearoff  (David Rubin+, July 2008) 
N-roll vs n-roll bearoff  (Gregg Cattanach, Nov 2002) 
N-roll vs n-roll bearoff  (Chuck Bower+, Dec 1997) 
Near end of game  (Daniel Murphy, Mar 1997) 
Near end of game  (David Montgomery, Feb 1997) 
Near end of game  (Ron Karr, Feb 1997) 
One checker model  (Kit Woolsey+, Feb 1998) 
Pip count percentage  (Jeff Mogath+, Feb 2001) 
Pip-count formulas  (Tom Keith+, June 2004)  [GammOnLine forum]
Thorp count  (Chuck Bower, Jan 1997) 
Thorp count  (Simon Woodhead, Sept 1991) 
Thorp count questions  (Chuck Bower, Sept 1999) 
Value of a pip  (Tom Keith, June 2004) 
Ward's racing formula  (Marty Storer, Jan 1992) 
What's your favorite formula?  (Timothy Chow+, Aug 2012) 

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